ACKNOWLEDGEMENTS
The author is indebted to Professor Frank Press for his support and guidance throughout this study. The micro- seism recordings in Tulsa, Oklahoma were made by the Jersey Production Research Company. Grateful acknowledgement is given to the Company and to Mr. F. G. Boucher for giving us the data and the seismic velocity information. Jn China Lake the recording was carried within the Naval Ordnance Test Station. We are grateful to Capt. Blenman, Jr., the Commander, NOTS, for granting permission and to Drs. P.
St. Amand and K. Odencratz, and to Mr. R. Zbur for their valuable assistance while working within the Station boundaries.
Mr. N. Motta1s most valuable help in design and building of instruments for microseism recording Is ack- nowledged. The author used the Fourier analys is and least squares phase velocity computer programs written by Mr. S. S~
Alexander. Valuable discussions were held with Drs.
s.
W.Smith, A. Ben-Menahem,
w.
L. Pilant, Mr.c.
B. Archambeau, and Mr. S. S. Alexander. Mr.w.
Taft and Mr. J. Romo were of great assistance in field work.This research was supported by Grant No. AF-AFOSR-25-63 of the Air Force Office of Scientific Research as part of the Advanced Research Projects Agency Project VELA, and by the National Science Foundation.
The figure preparations were supervised by Mr. L.
Len c he s , and t he aut h or w i s h e s to ex t end s p e c i a 1 t hanks to him.
ABSTRACT
A study of microseisms is made to determine some of their statistical properties and to investigate the feasibi 1 ity of their use in determining the shallow structures of the earth's crust by the phase velocity method. It is found that the microseisms in the period
I
range of to 6 seconds arrive from several directions with comparable strength and at the same time. There are occas-
ional short intervals of 10 - 40 seconds during which micro- seisms are mostly unidirectional. It is also found that these relatively short period microseisms are not stationary
in the wide sense over time intervals longer than 5 or 10 minutes.
The phase velocities of microseisms recorded with an array of 8 instruments are measured In four different
locations. The velocities, although scattered, are found to be in agreement with the theoretical dispersion curve for the fundamental Rayleigh mode, computed using the
available seismic velocity information. An error analysis is made and the confidence 1 imits are placed within ±20 percent of the measured velocities.
TABLE OF CONTENTS
Page INJRODUCTION . . • . .
. . . . . . . . . .
STATISTICAL PROPERTIES AND DIRECTION OF APPROACH
OF MICROSEISMS . • . . . • • • . . . • • . . . 5 Statistical Properties • • • • • • •
Direction of Approach of Microseisms • • • • I A METHOD FOR MEASURING THE PHASE VELOCITY OF
5 9
MICROSEISMS • . • . • • . . • • • . • • • 13 Instrumentation . • • .
Field Procedure . • . • • Analysis of Data • • • • • . RESULTS FOR DIFFERENT REGIONS
SOURCES OF ERROR AND RELIABILITY OF
MEASUREMENTS . • • • . • . . . • . .
14 16 17 21
29 Interference . • • • • • • • • • • • • • 29 Measurement Errors • • • • • • . • • • • 33 Rel iabi 1 ity of the Results • • • 39 CONCLUSIONS
REFERENCES . . LIST OF TABLES TABLES
FIGURE CAPTIONS ILLUSTRATIONS APPEND I X . . .
Time Series Analysis
40
44 48 49
51 54
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INTRODUCTION
In most earlier applications of the dispersive
properties of surface waves to crustal studies, waves with periods longer than 10 seconds were used, and the near surface properties of the crust were ignored. The
1know- ledge of the very shallow structure of the crust in local- ized areas carries great geologic significance. It is only in a very small fraction of the continents that the thickn~s
of the sedimentary rocks have been investigated by gravity and by seismic reflection and refraction methods. The surface wave dispersion method could be used in such are~s as well as the igneous and metamorphic regions for similar
investigations, provided waves with short wavelengths can be recorded and their phase velocities can be measured.
One source of short period surface waves are micro- seisms. Although these waves are more complicated than the transient surface waves generated by earthquakes and
explosions, their use is advantageous because they are universal and always present. This project was undertaken to evaluate the feasibility of measuring the phase velociti~
using a small multi-channel array of matched vertical seis- mometers, and determining the shallow structures from the observed phase velocity curves.
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Microseisms have been investigated by seismologists since the advent of the science~ and some papers on the subject were published as early as 1874 (de Rossi, 1874) . Most of the studies in this field are directed toward the
clarification of three major points: (1) The origin of microseisms, (2) nature of the microseismic waves~1 mode of pro p a gat i on, and d i r e c t i o n of approach , ( 3 ) some s tat i s t i ca 1 properties of microseisms and their treatment as noise.
There are several well known theories of origin of
microseisms~ but no one theory completely accounts for all the observations (references are listed in Gutenberg and Andrews, 1952~ 1956; Gutenberg~ 1958; Haubrich and lyer~
1962). It has been general ly accepted that microseisms originate in the oceans or in great lakes. In this general-
ization the high frequency vibrations due to wind and industrial noise are excluded. Microseismic waves are of both Rayleigh and Love type with the Rayleigh waves being more common (Ramirez~ 1940; Wilson~ 1942; Blaik and Donn~
1954; Darbyshire, 1954; Deacon~ 1954; Gutenberg~ 1958; Jensen~ 1958; B~th, 1962; lyer, 1962). The periods of the waves extend from 0.2 second to 30 seconds (Oliver~ 1962)~
with the most commonly observed spectral band being from 1 to 10 seconds. The waves are mostly of the fundamental mode with some higher modes at shorter periods (Gutenberg~
1958; Archambeau and Alexander, 1963). The direction of
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approach of the microseisms varies with time, and usually the waves arrive in more than one direction at a given time (Kishinouye, 1947; Leet, 1949, 1950; Ramirez, 1953;
Blaik and Donn, 1954; Donn, 1954; Gutenberg, 1958; Okano, 1961; Haubrich, Munk, and Snodgrass, 1963). The multi- directionality is due to extended and numerous sounces and to lateral refraction and reflections (Blaik, and Donn, 1954; Donn, 1954). Microseismic waves are attenuated when crossing geologic discontinuities. Also, Rayleigh-to-Love conversion has been observed over the discontinuities
( G u t e n b e r g , 1 95 8; R y k u n o v an d M i s h i n , 1 96 1 ) .
With the interest In seismic noise and noise el im~
nation, the statistical properties of microseisms have b e come i m p o r t an t i n r e c en t year s ( S p i e k e r , 1 96 1 ; H au b r i c h and lyer, 1962). Not enough work has been done in this field, however, to draw general conclusions.
A few attempts have been made for measuring the phase velocity of microseisms using a tripartite method (Ramirez, 1940; Mukherjee, 1948; Dinger, 1951; Lynch, 1951; Gutenberg, 1958; Okano, 1961; Rykunov, 1961). In most cases the
measured velocities scattered greatly and were of no physi~l
significance. In some other cases the average of the
measured velocities was too high for the particular structure.
In these measurements, however, the multi-directionality of
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the waves and the fact that the tripartite array could be used only when the waves were unidirectional were not taken
into account.
Before choosing the method of microseism phase velocity measurement, it was necessary to investigate the statistical properties and the multi -directionality In
d eta i 1 i n t h e per i o d range of our a p p 1 i cat i on ( 1 to 6 s &con ds ).
These investigations will be discussed briefly in the next section, then, the instruments designed for recording the microseisms in the field, the techniques used for phase velocity measurement, and the results obtained for four different areas will be described. Discussion of the in er- ference effects, numerical sources of error, and the
reliability of the measured phase velocities will precede the conclusions.
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STATISTICAL PROPERTIES AND DIRECTION OF APPROACH OF MICROSEISMS
Statistical Properties
The investigation of the statistical properties of mlcroseisms were not begun until recent years due to the
I
fact that such an investigation requires great amounts of data in digital form or in analog form·suitable for analysis.
The first continuous digital recording of microseisms along with the correlation and cross-spectra results are des-
cribed by Haubrich and lyer (1962). Another group (Spieker, 1961) investigated the stationarity of microseisms in time and space. The latter study shows that, in general, micro- seisms recorded at a given station are not stationary over a time interval of one hour. In other words, the auto-
correlation function is not invariant over such a time span.
This implies that in any kind of analysis where stationarity is assumed time length cannot exceed one hour. In space, it was found that waves were "almost" stationary over a dis-
tance of one kilometer, where stationarity in this case refers to invariance of the time autocorrelation function from one station to another at the same time.
Since it is advisable to know the stationarity prop- erties before the microseisms could be subjected to con- ventlonal spectral analysis, such a study was carried out
~-
using the microseisms recorded in Pasadena by the Caltech digital seismograph. A series of autocorrelation functions of microseisms recorded on two different days were computed using 2 minute long records. Before the correlation the time series were filtered. The microseism spectrum covers a wide frequency band extending from about 0.03 cpsl to 10
(or higher) cps. One would not expect the properties to be the same over the entire frequency range, since microseisms within different bands are associated with different sources.
The filtering was done to pass the waves in the period range of our interest, 0.5 to 5.0 seconds. The autocorre- lation functions Rt(T) were computed using record segments starting at t = 0, 3, 6, 13, 20, and 27 minutes. These are shown in figure 1 for two different days: September 18, 1962 and January 30, 1963.
Now, let us compare the autocorrelation functions for different origin times. If microseisms were stationary in the general sense, the autocorrelation functions would be
identical. In looking at the January 30th case, one sees that from t = 0 tot= 3 minutes Rt(T) changes some, but peak-to-peak correspondence is good. From t = 3 to t = 6, the change is more obvious. Rt(T) at t = 6 and t = 13 are very much alike. So is the case for t = 20 and t = 27. Bu~
from t = 6 tot= 20 minutes, there is a very definite change in the shape of Rt(T).
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For September 18th microseisms~ one observes a more systematic change in Rt(T) as t increases from t = 0 to t
=
3 and t=
6 minutes. For the 7 minute jumps in the origin times the variations in Rt(T) are outstanding. From these examples it can be concluded that in this particular period range (0.5 to 5 seconds) the microseisms appear to be stationary for time intervals less than about 10 minutes.For longer time durations they are not stationary.
The implications ofnonstationarity is that the
correlation functions and power spectra are time dependent~
and cannot be treated simply as functions of frequency.
Depending on the rate of time variation~ the correlation functions have to be computed over relatively short time intervals and the power spectra have to be obtained by
taking the Fourier transform of these short time functions.
These time dependent spectra are call ed "instantaneous spectra" (Page~ 1952; Silverman~ 1957). They could be con- sidered the generalized spectra with the conventional~ time
independent spectra being a special case. In practice~
however~ the use of the instantaneous spectrum concept is limited to special cases.
One question may arise in regard to our stationarity test. The autocorrelation functions were computed using finite records of 2-minute durations. In the rigorous definition of the correlation functions of aperiodic time
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series, the 1 !mits are extended to infinity. In practice this is not possible, and one is limited to the finite record lengths. A process which might be stationary in real !ty, could be found to benonstationary in practice when finite time intervals are used in testing. It is useful from the practical point of view to define stationarity in
I
terms of the time durations pertinent to a given experiment, since infinitely long time intervals cannot be realized.
Another statistical property of microseismic waves that requires investigation is their variation in space. A measure of this variation can be obtained by computing the coherence between the simultaneous recordings at two stations.
Since microseisms are continuous wave trains, coherence R would be equal to one for unidirectional stationary arrivals.
Any deviation from R=l would be due to the interference of uncorrelated waves, and R would decrease with the increasing
interference. The coherence between recordings at two stations with varying distances was computed using a uni - directional, 45-second long section of the microseisms recorded in the Imperial Valley, California. The traces recorded simultaneously at five stations were digitized at the rate of 10 samples per second. The digital data were detrended and filtered with a low-pass digital fi lter with cut-off at one second. Power spectra were obtained from the Fourier transform of the two-sided covariance functions, and
-9-
the coherence between pairs of stations was computed using the definition given in the Appendix. The results for five distances are shown in figure 2. Over a distance of 1.5 km the waves are coherent although there is a very slight
decrease in coherence with increasing distance. This decrease is more pronounced at shorter periods.
The spatial coherency of the waves is a very impor- tant test for the feasibi 1 ity of the phase velocity method . Unless the waves can be correlated from one station to
another, they cannot be used for phase velocity measure- ments . Figure 2 illustrates the excellent interstation coherence of the recorded microseisms over the maximum
dimensi ons of the array. For periods longer than 2 seconds the coherence is always larger than 0.8.
Direction of Approach of Microseisms
Several methods are used for determining the direc- tion of approach of microseisms. Three methods using the horizontal and vertical components of the motion are
described by Bgth (1962) . The tripartite method utilizes the same component of the motion at three stations. The
"cross-spectrum" method utilizes the complex cross-spectral components of the three-component station to give the direc- tion of arrival . The azimuthal angle 8 is given by
- 1 ONz
8
=
tan Q ( 1 )EZ
-10-
where N, E, Z, refer to north, east, and vertical components, and Q· 0
I J is the imaginary part of the complex spectrum Sij = Cij + iQij. A parameter characteristic of the beam width is (Haubrich, Munk, and Snodgrass, 1963).
2 2
ONz
+ QEZ 8 = ----~~--~~--Czz (CNN + CEE)
(2)
For Rayleigh waves approaching from a single direction 8=1, and for waves arriving with uniform density from every
direction 8=0. In general the beam width is found to be less than 8=0.5 (Haubrich, Munk, and Snodgrass, 1963) indi- eating that the direction of arrival is not unique.
All the above methods of determining the direction of microseisms would work as long as the basic assumption that the microseisms consist of unidirectional Rayleigh waves, or uncorrelated Rayleigh and Love waves would hold.
In the absence of these conditions, which in general is the case, the direction determined by any one of these methods is some kind of an average which has no physical significance.
A method which is most suitable for the study of microseism direction is the particle trajectory method.
For pure unidirectional, fundamental mode Rayleigh waves the two horizontal, north and east, components of the motion are linearly related, while, in general, the particle motion traces a retrograde ell ipse in the vertical plane. Two
examples of such motion for microseisms are shown in
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figures 3a, 3~ 3c. The data was taken from the Caltech digital seismograph and narrow-band filtered around the period T = 4 second to minimize the interference of other frequencies. All these figures show that in these cases the particle motions in the vertical plane are undisturbed retrograde ellipses. Hence, these waves consist of 1uni- directional fundamental mode Rayleigh waves. The direction can be computed with an accuracy of better than ± 5 degrees.
Figure 4 illustrates the case where the wave is not a uni- directional, pure Love or pure Rayleigh wave. Figures Sa and 5b show two cases where the direction of the interfered wave changes by about 90 degrees within 11 and 25 seconds,
respectively. The effectiveness and the accuracy of the direction from orbital motion can be illustrated with the
identification of the P and the SH waves from a small tremor during a strong microseismic storm on January 30, 1963.
Figure 6 shows the 1 inear relations between theN-Sand E-W components and the rotation of the 1 ine of polarization by exactly 90 degrees from P to SH. The earthquake was so
small (M~ 3) that it was recorded on the digital seismograph which was running at a very high gain, and was not visible above the noise level on photographic recordings in Pasaden~
Okano (1961) carried out a similar investigation of micro- seism motion using a vector seismograph. His conclusions
-12-
also support the rapid interference and direction changes of microseismic waves.
The rapid changes of direction introduce the most serious difficulty in the measurement of phase velocity of microseisms. The effect of interference on the observed phase velocity of the waves is derived later for special cases. When interfered, the waves are modulated in space, and if a simple tripartite array were used for phase velocizy measurement, in general it would not be possible to correlate the peaks from one station to another. If the phase dif- ferences were measured from Fourier phase spectra or cross- spectra, the results would have no phys ical significance since the spectra cannot be written as the product of a
space independent amplitude factor with an exponential phase factor. In previous phase velocity measurements these compli- cations were ignored, and as a result no reasonable phase velocity curve was obtained for microseisms. A typical example of such an effect is illustrated by Okano (1961, figure 12) where the phase velocities of 3 to 5 second microseisms are uniformly scattered between 1.0 km/sec and 3.0 km/sec.
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A METHOD FOR MEASURING PHASE VELOCITY OF MICROSEISMS A method that is to be used for measuring phase
velocity of microseisms must have the following properties:
1. It must work with time records 20 - 30 seconds long, 2. it must have some provision for identifying interfered
I
and pure unidirectional wave trains, 3. it must have enough accuracy for measurement of phase velocity over small arrays.
The 1 imitation of record length arises from the fact that it is only possible to find short segments of the
record where the microselsms are unidirectional. Since
cross-spectra cannot be used with such short record lengths, Fourier phase spectra and direct time delay measurements have to be utilized between stations. The second require- ment is to assure an uninterfered wave train regardless of
length, and it can be realized by using a close array of stations to follow the progress of the wave train. The restriction on the maximum size of the array is due to the fact that shallow structures of the earth's crust may
change rapidly, and the array must be small to measure
velocities over limited areas. To meet these qualifications special instruments were built, and phase velocities were measured at four different locations.
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Instrumentation
A set of 8-channel portable instruments were designed and built for field recording of mlcroseisms. In designing the instruments the author placed emphasis on matching the phase response of the system rather than shaping the ampl i- tude response curve. ~herwise instrumental effect~ would mask the phase difference of the signal, which over a short array is only a small fraction of a circle. The seismometers used were modified, one second, variable
reluctance, portable Benioff instruments. The periods were made adjustable by using an external suspension system and varying the axes of the suspending negative length springs from the vertical . The maximum deviation between the
seismometer periods was kept less than 2 percent of the mean period. The signal from seismometers was transmitted to a test panel in the recording trailer using seismic cables. The seismometers were run at critical damping
where the damping resistance, taking into account the cable resistance, was adjusted at the input of the amplifiers.
The amplification was done by transistorized, double- loop, D-C amplifiers with a maximum voltage gain of 200,000. A low-pass R-C filter unit with three different roll-off frequencies and slopes of either -12 or ~8 db/octave was inserted between the two stages of the amplifier. In